INTENSITY OF ORDINARY AND EXTRAORDINARY RAYS. 409 
Thus the equations, on account of the complete transparenc 
q I I Ys 
become— 
1=[tan?22° 30! + tan? 2° 32']s? + o? tan? 67° 32! 4 o!? tan? 24° 35/, 
1=[1 + tan? 2° 31']s?4 02+ ol, 
O= [tan 22° 30! . tan 2° 31'— tan 2° 32']s?—o” tan 67° 32!. 
4+ o!2 tan 24° 35!. 
Hence s?=0°'1078. o'?=0°1409. 
p?=0'01850. w'?=0°8241. 
s'2=0-000207. o"2—07510. 
p'?=0°000212. w2=(15 72, 
By means of these values we now find the intensity of the 
reflected ray and that of the two refracted ones, in whatever azi- 
muth the incident ray is polarized ; thus we have only to substi- 
tute these values in the expressions for R,, R,, D', D". If the in- 
cident light is common and its intensity = 1, the intensity of the 
2 2 12 12 
reflected ray is SESS TE = 0°0633 ; that of the ordinary 
is 0°4825 ; of the extraordinary 0°4541. In my treatise which I 
just now alluded to, I have calculated these three intensities and 
find them 0:0632 for the reflected light, 0°4825 for the ordinary 
ray, and 0°4542 for the extraordinary ray; so that they agree 
almost perfectly to the fourth decimal place. An accurate dis- 
cussion of the errors which can possibly occur in the final result 
would show that, by this method of estimating the intensity, 
the quantities of reflected and refracted light may be accurately 
calculated even to ;,,th of that incident. 
In a third letter* to M. Ampére, M. Cauchy has reproduced 
Fresnel’s formula for total reflexion from his new principles. The 
principles of my theory, applied to this case, give the same ex- 
pressions. I was first led to its application from an observation 
in M. Cauchy’s paper on the use of experimental magnitudes 
instead of the imaginary sines and cosines, hence they are not 
used in my treatise, but I will take the opportunity of com- 
municating them here. The differential equations, upon which 
the movements of the zther in an uncrystalline transparent me- 
dium depend, when the plane of vibration is parallel to the co- 
ordinate axis z, and uw, v, w express the movements of the par- 
ticles parallel to the axis 2, y, z, are,— 
* Comptes Rendus, lier Sem. 1836, p. 386+. 
VOL. IV. PART XY. 2G 
